Elaborate on operations, linear lambda.

biblio.bib

@@ -7,3 +7,15 @@
   organization={Springer}
 }
 
+@article{castellan2016concurrent,
+  title={Concurrent Games},
+  author={Castellan, Simon and Clairambault, Pierre and Rideau, Silvain and Winskel, Glynn},
+  journal={arXiv preprint arXiv:1604.04390},
+  year={2016}
+}
+@book{milner1980ccs,
+  title={A calculus of communicating systems},
+  author={Milner, Robin},
+  year={1980},
+  isbn={0387102353}
+}

concurgames.sty

@@ -2,6 +2,8 @@
 %% Useful commands for concurrent games as event structures
 
 \usepackage[normalem]{ulem}
+\usepackage{MnSymbol}
+\usepackage{stmaryrd}
 
 \newcommand{\con}{\operatorname{Con}}
 \newcommand{\confl}{\raisebox{0.5em}{\uwave{\hspace{2em}}}}
@@ -9,9 +11,12 @@
 \newcommand{\forkover}[1]{[\qtodo{fork}~#1]}
 \newcommand{\fork}{[\qtodo{fork}]}
 \newcommand{\edgeArrow}{\rightarrowtriangle}
+\newcommand{\linArrow}{\rightspoon}
 
 \newcommand{\config}{\mathscr{C}}
 
 \newcommand{\strComp}{\odot}
 \newcommand{\strInteract}{\ostar}
 \newcommand{\strParallel}{\parallel}
+
+\newcommand{\seman}[1]{\llbracket{} #1 \rrbracket}

report.tex

@@ -12,10 +12,10 @@
 
 % Custom packages
 \usepackage{leftrule_theorems}
-\usepackage{concurgames}
 \usepackage{my_listings}
 \usepackage{my_hyperref}
 \usepackage{math}
+\usepackage{concurgames}
 
 \newcommand{\qtodo}[1]{\colorbox{orange}{\textcolor{blue}{#1}}}
 \newcommand{\todo}[1]{\colorbox{orange}{\qtodo{\textbf{TODO:} #1}}}
@@ -176,6 +176,103 @@ and not just a set where the events from, \eg, $A$ and $B$ are mixed. \\
 This information is kept in a tree, whose leaves are the base games that were
 put in parallel, and whose nodes represent a parallel composition operation.
 
+Finally, a \emph{strategy} is consists in a game and an ESP (the strategy
+itself), plus a map from the nodes of the strategy to the nodes of the game.
+This structure is really close to the mathematical definition of a strategy,
+and yet is only a lesser loss in efficiency.
+
+\subsection{Operations}
+
+The usual operations on games and strategies, namely \emph{parallel
+composition}, \emph{pullback}, \emph{interaction} and \emph{composition} are
+implemented in a very modular way: each operation is implemented in a functor,
+whose arguments are the other operations it makes use of, each coming with its
+signature. Thus, one can simply \lstocaml{open Operations.Canonical} to use the
+canonical implementation, or define its own implementation, build it into an
+\lstocaml{Operations} module (which has only a few lines of code) and then
+open it. This is totally transparent to the user, who can use the same infix
+operators.
+
+\subsubsection{Parallel composition}
+
+While the usual construction (\cite{castellan2016concurrent}) involves defining
+the events of $A \strParallel B$ as ${\set{0} \times A} \cup {\set{1}
+\times B}$, the parallel composition of two strategies is here simply
+represented as the union of both event structures, while altering the
+composition tree.
+
+\subsubsection{Pullback}
+
+Given two strategies on the same game, the pullback operation attempts to
+extract a ``common part'' of those two strategies. Intuitively, the pullback of
+two strategies is ``what happens'' if those two strategies play together.
+
+The approach that was implemented (and that is used as
+\lstocaml{Pullback.Canonical}) is a \emph{bottom-up} approach: iteratively, the
+algorithm looks for an event that has no dependencies in both strategies, adds
+it and removes the satisfied dependencies.\\
+One could also imagine a \emph{top-bottom} approach, where the algorithm starts
+working on the merged events of both strategies, then looks for causal loops
+and removes every event involved.
+
+\subsubsection{Interaction}
+
+Once the previous operations are implemented, \emph{interaction} is easily
+defined as in the literature (\cite{castellan2016concurrent}) and nearly is a
+one-liner.
+
+\subsubsection{Composition}
+
+Composition is also quite easy to implement, given the previous operations. The
+only difficulty is that hiding the central part means computing the new
+$\edgeArrow$ relation (that is the transitive reduction of $\leq$), which means
+computing the transitive closure of the interaction, hiding the central part
+and then computing the transitive reduction of the DAG\@.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Linear lambda-calculus}
+
+Concurrent games can be used as a model of lambda-calculus \todo{ref Pierre~?}.
+To avoid non-determinism in the strategies, and to have a somehow easier
+approach, one can use concurrent games as a model of \emph{linear}
+lambda-calculus, that is, a variant of the simply-typed lambda-calculus where
+each variable in the environment can and must be used exactly once.
+
+\begin{minipage}{0.4\textwidth} \begin{equation}
+	\tag{\textit{Ax}}
+	\frac{}{x : A \vdash x : A}
+	\label{typ:llam:ax}
+\end{equation} \end{minipage}
+\hfill
+\begin{minipage}{0.5\textwidth} \begin{align}
+	\tag{\textit{App}}
+	\frac{\Gamma \vdash t : A \linArrow B \quad
+	\Delta \vdash u : A}
+		{\Gamma,\Delta \vdash t~u : B}
+		(\Gamma \cap \Delta = \emptyset)
+	\label{typ:llam:app}
+\end{align} \end{minipage}
+
+\todo{describe linear lambda-calculus?}
+
+The implementation, which was supposed to be fairly simple, turned out to be
+not as trivial as expected due to technical details: while, in the theory, the
+parallel composition is obviously associative and commutative (up to
+isomorphism), and thus used as such when dealing with environment and typing
+rules, things get a bit harder in practice when one is supposed to provide the
+exact strategy.
+
+For instance, the above rule~(\ref{typ:llam:app}) states that the resulting
+environment is $\Gamma,\Delta$, while doing so in the actual implementation
+(that is, simply considering $\seman{\Gamma} \strParallel \seman{\Delta}$)
+turns out to be a nightmare: it is better to keep the environment ordered by
+the variables introduction order, thus intertwining $\Gamma$ and $\Delta$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Linear CCS}
+
+\cite{milner1980ccs}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \bibliography{biblio}